Abstract In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in three spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has a complex spatial variation, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does.Abstract In Bohmian mechanics, the nodes of the wave function play an important role in the generation of chaos. However, so far, most of the attention has been on moving nodes; little is known about the possibility of chaos in the case of stationary nodes. We address this question by considering stationary states, which provide the simplest examples of wave functions with stationary nodes. We provide examples of stationary wave functions for which there is chaos, as demonstrated by numerical computations, for one particle moving in three spatial dimensions and for two and three entangled particles in two dimensions. Our conclusion is that the motion of the nodes is not necessary for the generation of chaos. What is important is the overall complexity of the wave function. That is, if the wave function, or rather its phase, has a complex spatial variation, it will lead to complex Bohmian trajectories and hence to chaos. Another aspect of our work concerns the average Lyapunov exponent, which quantifies the overall amount of chaos. Since it is very hard to evaluate the average Lyapunov exponent analytically, which is often computed numerically, it is useful to have simple quantities that agree well with the average Lyapunov exponent. We investigate possible correlations with quantities such as the participation ratio and different measures of entanglement, for different systems and different families of stationary wave functions. We find that these quantities often tend to correlate to the amount of chaos. However, the correlation is not perfect, because, in particular, these measures do not depend on the form of the basis states used to expand the wave function, while the amount of chaos does.

arXiv:1603.01387Journal of Physics A Mathematical and TheoreticalSep 2, 201617 citations

Impact of dark states on the stationary properties of quantum particles with off-centered interactions in one dimension

G. Bougas, Nathan L. Harshman, Peter Schmelcher

We present a generalization of the two-body contact interaction for nonrelativistic particles trapped in one dimension. The particles interact only when they are a distance $c$ apart. The competition of the interaction length scale with the oscillator length leads to three regimes identified from the energy spectra. When $c$ is less than the oscillator length, particles avoid each other, whereas in the opposite case bunching occurs. In the intermediate region where the oscillator length is comparable to $c$ , both exclusion and bunching are manifested. All of these regions are separated by dark states, i.e., bosonic or fermionic states which are not affected by the interactions.We present a generalization of the two-body contact interaction for nonrelativistic particles trapped in one dimension. The particles interact only when they are a distance $c$ apart. The competition of the interaction length scale with the oscillator length leads to three regimes identified from the energy spectra. When $c$ is less than the oscillator length, particles avoid each other, whereas in the opposite case bunching occurs. In the intermediate region where the oscillator length is comparable to $c$, both exclusion and bunching are manifested. All of these regions are separated by dark states, i.e., bosonic or fermionic states which are not affected by the interactions.

arXiv:2403.10078Physical Review AAug 27, 20240 citations

Particle in One Dimension

‎Louis Marchildon

Springer eBooksJan 1, 20020 citations

Existence Conditions of Particle's Bound State in One-dimensional Unfinite Deep Potential Well

Le Wang

For a moving particle in an semi- infinite potential well of one dimension,a continuous and monotonous function is introduced by solving stationary Schr? dinger equation,then Existence conditions of one even more Particle's bound state are analyzed by means of the zero- point theorem of continuous function. The results show the number of bound state is connected with the width and depth of one – dimensional Infinite potential well. For a certain quality of particle,more bound states need much deeper and wider potential well.For a moving particle in an semi- infinite potential well of one dimension,a continuous and monotonous function is introduced by solving stationary Schr? dinger equation,then Existence conditions of one even more Particle's bound state are analyzed by means of the zero- point theorem of continuous function. The results show the number of bound state is connected with the width and depth of one – dimensional Infinite potential well. For a certain quality of particle,more bound states need much deeper and wider potential well.

Journal of Heze UniversityJan 1, 20142 citations

We present a generalization of the two-body contact interaction for nonrelativistic particles trapped in one dimension. The particles interact only when they are a distance

c

apart. The competition of the interaction length scale with the oscillator length leads to three regimes identified from the energy spectra. When

c

is less than the oscillator length, particles avoid each other, whereas in the opposite case bunching occurs. In the intermediate region where the oscillator length is comparable to

c

, both exclusion and bunching are manifested. All of these regions are separated by dark states, i.e., bosonic or fermionic states which are not affected by the interactions.